Inverse Laplace Transform of (2s + 3)/((s + 1)(s² + 4))
The inverse Laplace transform of (2s + 3)/((s + 1)(s² + 4)) is 1/5·e^(-t) - 1/5·cos(2t) + 11/10·sin(2t). See the partial fraction expansion and the transform pair used for each term, then edit F(s) to try your own.
L⁻¹{(2s + 3)/((s + 1)(s² + 4))} = 1/5·e^(-t) - 1/5·cos(2t) + 11/10·sin(2t). A mixed decomposition: one real pole plus a conjugate pair.
e.g. 2s+3
Expanded or factored, e.g. (s+1)(s^2+4)
The result
L⁻¹{(2s + 3)/((s + 1)(s² + 4))} = 1/5·e^(-t) - 1/5·cos(2t) + 11/10·sin(2t)
Partial fractions: (1/5)/(s + 1) + (-(1/5)s + 11/5)/(s^2 + 4)
- Invert (1/5)/(s + 1) — L⁻¹{A/(s − a)^1} = A·t^0e^(at)/0!
- Invert (-(1/5)s + 11/5)/(s^2 + 4) — L⁻¹{(Bs + C)/((s − a)² + b²)} = e^(at)[B·cos bt + ((C + aB)/b)·sin bt]
A mixed decomposition: one real pole plus a conjugate pair.
More inverse transforms
Frequently asked questions
What is the inverse Laplace transform of (2s + 3)/((s + 1)(s² + 4))? ▾
It is f(t) = 1/5·e^(-t) - 1/5·cos(2t) + 11/10·sin(2t). A mixed decomposition: one real pole plus a conjugate pair.
What are the partial fractions here? ▾
F(s) splits into (1/5)/(s + 1) + (-(1/5)s + 11/5)/(s^2 + 4), and each piece is inverted separately.
How can I check this answer? ▾
Transform 1/5·e^(-t) - 1/5·cos(2t) + 11/10·sin(2t) forward again with the Laplace transform calculator — you should get back (2s + 3)/((s + 1)(s² + 4)).
About this Calculator
The inverse Laplace transform of (2s + 3)/((s + 1)(s² + 4)) is 1/5·e^(-t) - 1/5·cos(2t) + 11/10·sin(2t). See the partial fraction expansion and the transform pair used for each term, then edit F(s) to try your own.